Equal Sign Research Articles

EXPLORING LEARNING OPPORTUNITIES FOR PRIMARY TEACHERS: THE CASE OF KNOWLEDGE FOR TEACHING EARLY ALGEBRA

Understanding how to constitute and develop opportunities for primary teachers teach early algebra to younger children is still an important research gap. In this paper, we bring results of a research program developed in Brazil over the past five years. We aim to discuss how professional learning opportunities emerged when teachers collectively planned, discussed, and analyzed lessons involving different meanings of the equality symbol and the development of functional thinking. Developed from the perspective of a qualitative-interpretative research, data analyzed consists of curriculum documents, protocols for the resolution of formative tasks, audios and videos collected during teacher education processes for in-service teachers. The results highlight that professional learning tasks, combined with the actions of teacher educators during collective discussions, favored in-service teachers to differentiate and understand the students’ reasoning. Some implications for teacher education as well as the professional development of primary teachers are discussed, especially regarding early algebra thinking, because teachers do not normally have the opportunity to study these contents on their own experiences in school.

Self-correlated spatial random variables: From an auto- to a sui- model respecification

This paper marks the 50-year publication anniversary of Besag's seminal spatial auto- models paper. His classic article synthesizes generic autoregressive specifications (i.e., a response variable appears on both sides of a regression equation and/or probability function equal sign) for the following six popular random variables: normal, logistic (i.e., Bernoulli), binomial, Poisson, exponential, and gamma. Besag dismisses these last two while recognizing failures of both as well as the more scientifically critical counts-oriented auto-Poisson. His initially unsuccessful subsequent work first attempted to repair them (e.g., pseudo-likelihood estimation), and then successfully revise them within the context of mixed models, formulating a spatially structured random effects term that effectively and efficiently absorbs and accounts for spatial autocorrelation in geospatial data. One remaining weakness of all but the auto-normal is a need to resort to Markov chain Monte Carlo (MCMC) techniques for legitimate estimation purposes. Recently, Griffith succeeded in devising an innovative uniform distribution genre—sui-uniform random variables—that accommodates spatial autocorrelation, too. Its most appealing feature is that, by applying two powerful mathematical statistical theorems (i.e., the probability integral transform, and the quantile function), it redeems Besag's auto- model failures. This paper details conversion of Besag's initial six modified variates, exemplifying them with both simulation experiments and publicly accessible real-world georeferenced data. The principal outcome is valuable spatial statistical advancements, with special reference to Moran eigenvector spatial filtering.

Abstract word comprehension in sentences for primary school readers: its effects on first graders with different levels of equal sign knowledge proficiency

Abstract word comprehension in sentences for primary school readers: its effects on first graders with different levels of equal sign knowledge proficiency

Development of prospective elementary teachers’ knowledge to teach early algebra through case discussions

AbstractThis study investigated the development of prospective elementary teachers (PETs)’ subject matter and pedagogical content knowledge for teaching early algebra through participation in case discussions. PETs attended five weeks of intervention as part of a method course, which included case discussions. During the intervention, the participants were presented with big ideas of equivalence and equations, generalized arithmetic, and functional thinking as the content of early algebra through text-based classroom cases. Data were collected through one-hour individual interviews before and after the early algebra lessons. The analysis of the pre-interviews indicated that prospective elementary teachers may not be ready to foster algebraic thinking in elementary grades in terms of the required knowledge for teaching. We found that PETs needed more development of required subject matter knowledge to guide algebraic thinking in elementary grades, such as the relational meaning of the equal sign, generalizing, representing, or justifying arithmetic or functional relationships, and reasoning with them. Likewise, PETs were found to lack in pedagogical content knowledge related to students’ conceptions and misconceptions and effective teaching strategies to promote early algebraic thinking. However, we found that specifically designed method courses could help develop PETs’ knowledge of teaching early algebra. Following early algebra lessons centered around case discussions, PETs showed progress in multiple aspects of knowledge across all big ideas, though their improvement in generalized arithmetic was somewhat less pronounced. This progress was observed in their subject matter and pedagogical content knowledge.

Semiotic mediation of gestures in the teaching of early algebra: the case of the equal sign

Semiotic mediation of gestures in the teaching of early algebra: the case of the equal sign

Algebraic Representations of a Splitting Task: Implications of Numerical Reasoning

ABSTRACT This qualitative research study uses middle-grades students’ numerical reasoning to model their symbolic representations of the relationship between two multiplicatively related unknowns on an algebra task. Students in sixth grade through ninth grade participated in clinical interviews that assessed their numerical reasoning using the Number Sequences framework, their interpretation of the equal sign, and their algebraic reasoning in the context of writing equations. Analysis used the mental structures that define their numerical reasoning to explain their algebraic capabilities, and changes in their reasoning about the equal sign in different contexts. Findings show that splitting, one mental structure related to the students’ numerical reasoning, is related to students’ symbolic representations of multiplicatively related unknowns. Furthermore, students who had not constructed a splitting operation tended to use numerical examples in place of reasoning about unknowns, and to reason operationally about the equal sign to compensate for this limitation on an algebra task.

Strong zero modes and edge states in the interacting fermion chain without pairing

The operator of the strong zero mode for the one-dimensional system of interacting fermions without pairing is presented. It is conjectured that the strong zero mode is related to the Majorana edge eigenstate, which is shown to exist (using the exact Bethe ansatz study) in this system. The results are robust with respect to the sign randomness of hopping amplitudes (and if the pairing amplitudes are nonzero, similar results exist for equal sign randomness of hopping and pairing amplitudes).

How Prior Knowledge, Gesture Instruction, and Interference After Instruction Interact to Influence Learning of Mathematical Equivalence.

Although children learn more when teachers gesture, it is not clear how gesture supports learning. Here, we sought to investigate the nature of the memory processes that underlie the observed benefits of gesture on lasting learning. We hypothesized that instruction with gesture might create memory representations that are particularly resistant to interference. We investigated this possibility in a classroom study with 402 second- and third-grade children. Participants received classroom-level instruction in mathematical equivalence using videos with or without accompanying gesture. After instruction, children solved problems that were either visually similar to the problems that were taught, and consistent with an operational interpretation of the equal sign (interference), or visually distinct from equivalence problems and without an equal sign (control) in order to assess the role of gesture in resisting interference after learning. Gesture facilitated learning, but the effects of gesture and interference varied depending on type of problem being solved and the strategies that children used to solve problems prior to instruction. Some children benefitted from gesture, while others did not. These findings have implications for understanding the mechanisms underlying the beneficial effect of gesture on mathematical learning, revealing that gesture does not work via a general mechanism like enhancing attention or engagement that would apply to children with all forms of prior knowledge.

The war on flags: The opposition to state-sponsored LGBTQ+ symbols

Negativity against LGBTQ+ and gender equality symbols is escalating across Western European countries, including those at the forefront of sexual modernism. Drawing on data from Spain, this paper theorizes and finds that state-sponsored LGBTQ+ symbols receive significantly more negativity than other aspects of LGBTQ+ issues related to general and specific attitudes toward formal rights. The negativity is primarily explained by support for Vox, a radical right-wing party, and age. The study provides insights into the complexities of public opinion surrounding LGBTQ+ symbols, offering a nuanced understanding of the challenges faced by the LGBTQ+ community in terms of acceptance and visibility. Furthermore, it highlights the influence of political affiliations and generational factors in shaping these attitudes.

Vietnam Cult of the Mother Goddess and its Influence on Confucian Ethics in Vietnam

Vietnam is a country with many spiritual beliefs that reflect the values of its inhabitants, being an important component of their traditional culture. A special place is occupied by faith in the Mother Goddess. This kind of beliefs, which is completely unique for Vietnam, has a long history and emphasizes the feminine principle through the image of a woman with the power and ability to create, enrich and develop everything that exists. Faith in the Mother Goddess reflects the values and high human qualities of the Vietnamese, and it has been recognized by UNESCO as an object of representative intangible cultural heritage of humanity. Confucian ethics was adopted in Vietnam, but it was influenced by the Vietnamese belief in the Mother Goddess. The cult of the Mother Goddess also had a great influence on the ideas about the role of women in society. The study analyzes the origins, features and values of this cult and indicates its influence on the Vietnamese Confucian ethics. It is established that the activities of the worship of the Mother Goddess contribute to strengthening the sense of community of the Vietnamese nation. In connection with the development of Confucian ethics within the framework of Vietnamese culture, the attitude towards women has changed to a more humane and open one. In modern times, the cult of mother worship has acquired secular features and can be seen as a sign of gender equality.

Cyclic inequalities for generalized mixed width-integrals

In this paper, we research the cyclic inequalities of generalized width-integral in convex geometry analysis. Based on previous studies, we use Hölder’s inequality and inverse Hölder’s inequality to study further the generalized width-integral inequalities of the convex body in Euclidean space, establish a series of cyclic inequalities, and give the condition of equality sign.

Run Length Limited Error Control Codes Derived from Reed Solomon Codes

Reed Solomon codes were standardized for numerous wireless communication systems. Most practical Reed Solomon codes belong to non-binary linear block codes defined over finite fields with characteristic two. Each linear block code contains one codeword composed of all zeros. The concatenation of this and also other codewords can lead to long, theoretically even infinite, runs of equal symbols. Such long runs do not support synchronization in wireless receivers and therefore are unwanted. In this paper it is shown that extended and some appropriately shortened Reed Solomon codes constructed over finite fields with characteristic two can be transformed into Run Length Limited Reed Solomon codes. The presented method, if applicable, allows for doing it without inserting additional redundancy. Another advantage is that after the transformation, if some round conditions are fulfilled, the decoding does not have to be rebuilt.

Concentration and condition of American lobster postlarvae in small‐scale convergences

AbstractInvertebrate larvae are often abundant in the surface ocean, which plays a key role in their dispersal and connectivity. Pelagic microhabitats characterized by small‐scale hydrographic variability are complex and ubiquitous in the coastal ocean, but their study is challenging, and they have been largely neglected in meroplankton ecology. Surface convergences, i.e., surface microhabitats featuring convergent horizontal currents, may aggregate the last larval stage of the American lobster and could provide shelter and food for Stage IV postlarvae and thus enhance their condition. We tested these hypotheses by conducting a series of cruises in the southwestern Gulf of Maine in summer 2021, sampling 15 paired sets of potential convergences and off‐convergence unstructured habitat. We measured postlarval abundance, surface hydrography, acoustic backscatter, and circulation. Experiments and image analysis compared condition, color, and morphology of postlarvae sampled inside and outside potential convergences. Potential convergences varied in near‐surface hydrographic patterns, with most displaying consistency among two transects and diverse patterns in salinity and temperature (e.g., across‐convergence gradients with equal or different signs). While the highest postlarval abundances were found in convergences, abundance patterns on and off convergences were not consistent, and another analysis indicated higher abundance in convergences than in a 7‐year untargeted surface ocean data set. Experiments indicated no survivorship differences among convergence and non‐convergence individuals at two temperatures, while image analyses revealed differences in color and size. Physical measurements and qualitative neuston community analyses indicated substantial heterogeneity among potential convergences. Our results reinforce that small‐scale heterogeneities are highly variable but important to the ecology of meroplankton, including the pelagic and neustonic habitats where lobster postlarvae are abundant.

Discrete quantum structures I: Quantum predicate logic

A discrete quantum structure is a discrete quantum space that is equipped with relations and functions of various finite arities. Discrete quantum spaces are identified with hereditarily atomic von Neumann algebras, their relations with projection operators, and their functions with unital normal * -homomorphisms. The propositional quantum logic of Birkhoff and von Neumann has been extended to a predicate quantum logic by Weaver; we investigate this predicate quantum logic as the internal logic of discrete quantum structures. We extend this predicate quantum logic to include function symbols and an equality symbol. Overall, we recover the basic structures of discrete quantum mathematics from physical first principles. More complicated structures will be recovered similarly in part II of this paper.

Developing the concept of task substitution and transformation by defining own equivalences

AbstractThe presented article is dedicated to a new way of teaching substitution in algebra. In order to effectively master the subject matter, it is necessary for students to perceive the equal sign equivalently, to learn to manipulate expressions as objects, and to perceive and use transformations based on defining their own equivalences. According to the results of several researches, these changes do not occur automatically, and the neglect of their development leads to students’ insufficient adoption of substitution. The submitted contribution presents a new way of teaching substitution, the stages of which support the gradual development of the necessary competences of students, so that substitution becomes part of their computing apparatus. The effectiveness of the mentioned method of teaching substitution was also verified experimentally. By conducting a pedagogical experiment, it was confirmed that the application of the substitution teaching method developed by us led to more frequent use of substitution by students from the experimental group (47 students) compared to students from the control group (82 students) who learned substitution in the usual way. It emerged from the interview with experimental group students that they considered the proposed method suitable and that it encouraged them to learn substitution in depth.

Learners’ Misconceptions and Errors when Solving Inequalities

Learners’ Misconceptions and Errors when Solving Inequalities

We Cannot Ignore the Signs: The Development of Equivalence and Arithmetic for Students from Grades 3 to 4

ABSTRACT Students' understanding of the meaning of the equal sign develops slowly over the primary grades. In addition to updating their representations of equations to recognize that the equal sign represents an equivalence relation rather than signaling an operation, students need to move beyond full computation to efficiently solve equivalence problems. In this study, we examined the longitudinal relation between arithmetic and equivalence for students who were capable of accurately solving arithmetic problems in different formats. Chinese students (N = 612; M age = 9.0 years in Grade 3, 57% boys) completed measures of arithmetic fluency and equivalence fluency in Grade 3 and again in Grade 4. They also completed a non-verbal reasoning task in Grade 3. We tested a cross-lagged structural equation model to examine the reciprocal relations between arithmetic and equivalence fluency. We found reciprocal relations between the development of arithmetic and equivalence fluency from Grades 3 to 4, with a greater influence of arithmetic on the development of equivalence than the reverse. Furthermore, non-verbal reasoning predicted the development of equivalence, but not the development of arithmetic. Based on our findings, we conclude that for Chinese students with prior basic understanding of equivalence, flexible access to arithmetic facts supports their development of equivalence fluency.

Kindergarten predictors of formal understanding of mathematical equivalence in second grade.

A longitudinal study was conducted to identify unique sources of individual differences in later understanding of the equal sign as a relational symbol of equivalence (i.e., formal understanding of mathematical equivalence). The sample included 141 children from a mid-sized city in the Midwestern United States (Mage = 6 years, 2 months in kindergarten; 88 boys, 53 girls; 71% white, 8% Hispanic or Latine, 7% Black, 3% Asian, 11% multiracial or other race/ethnicity; 42% qualified for free/reduced lunch). Children were assessed on three categories of skills in kindergarten including number knowledge, relational thinking, and executive functioning. These skills were hypothesized to provide a foundation for a formal understanding of mathematical equivalence (assessed in second grade) by preventing a specific, narrow misunderstanding of the equal sign that hinders learning. Results showed that kindergarten relational thinking, particularly tasks assessing nonsymbolic equivalence understanding, uniquely and positively predicted formal understanding of mathematical equivalence and negatively predicted the specific misunderstanding of the equal sign in second grade, controlling for IQ, gender, and free/reduced lunch status. Exploratory analyses unpacking the categories of skills into individual tasks also indicated specific areas of kindergarten instructional focus that may help children construct understanding of mathematical equivalence in future years. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

초등학생의 등호에 대한 의미 이해에 따른 등식 구조 이해 및 해결 전략 분석

The purpose of this study is to explore in what ways students' understanding of the equal sign affects their understanding of equation-structure and equation-solving. For this purpose, a questionnaire of the equal sign was developed, and a total of 450 students from the 20 fourth-grade classes were selected as research participants. Based on the results, we categorized students' understanding of the equal sign into three groups: sameness, answer, and substitution, and compared and contrasted their answer rates and solution strategies in equation-structure and equation-solving items. The overall results showed that the students of substitution, sameness, and answer were more successful in solving items and could employ more relational strategies. We also analyzed students' written definitions of the equal sign and found a need to refine them. Based on these findings, we discuss implications for how to address and teach the equal sign to elementary school students.

Eksplorasi Berpikir Aljabar Siswa Kelas 5 Dalam Menyelesaikan Soal Pemodelan

Algebraic thinking is important for developing students' mathematical generalization abilities, and identifying patterns, shapes, and symbols. This study uses a qualitative descriptive approach that aims to describe the profile of students' algebraic thinking in solving modeling problems. The subjects of this study were 158 fifth-grade students in two private schools in Surakarta, Central Java. The data collection instrument used was six test questions related to the modeling component adopted from Ralston. Prior to use, the instrument was validated by 3 elementary school mathematics learning experts and tested on 10 students who were not the subject of the study. Based on the test results, there were 12 students who scored in the high category. This study focuses on analyzing the algebraic thinking profile of students with high categories. The results showed that high-category subjects were able to demonstrate understanding related to modeling indicators in algebraic thinking, namely understanding the meaning of the equal sign (=) as an equivalence relationship, using variables to solve problems in the form of equations, and understanding the relationships between arithmetic operations. However, a small number of subjects still experience calculation errors and understanding errors in algebraic operations. Hence, it can be concluded that subjects with high categories are able to demonstrate algebraic thinking skills in the modeling component